The Gamma-Ray Burst ( GRB ) prompt emission is believed to be from highly relativistic electrons accelerated in relativistic shocks . From the GRB high-energy power-law spectral indices \beta observed by the Burst and Transient Source Experiment ( BATSE ) Large Area Detectors ( LAD ) , we determine the spectral index , p , of the electrons ’ energy distribution . Both the theoretical calculations and numerical simulations of the particle acceleration in relativistic shocks show that p has a universal value \approx 2.2 - 2.3 . We show that the observed distribution of p during GRBs is not consistent with a \delta -function distribution or an universal p value , with the width of the distribution \geq 0.54 . The distributions of p during X-ray afterglows are also investigated and found to be inconsistent with a \delta -function distribution . The p -distributions in blazars and pulsar wind nebulae are also broad , inconsistent with a \delta -function distribution .